Group #29579512 — subgroups

Subgroups — #29579512

Order 15616.

454 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840142	C₂	1	1	yes	maximal, characteristic
2	#21444492	C₂	1	4	yes	maximal
3	#21444111	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1837	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₄₄	—	2	2	no	—
6	C₂×#14519777	C₂²	1	2	yes	—
7	#16840142	—	2	2	no	—
8	#16840142	C₄	1	4	yes	—
9	#16839815	C₂²	1	2	yes	—
10	#16839509	—	2	8	no	—
11	#16839484	—	2	2	no	—
12	C₆₁×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#341	—	2	2	no	—
15	C₂²×Dic₁₂₂	—	2	2	no	—
16	C₂×Dic₂₄₄	—	2	8	no	—
17	C₂×Dic₂₄₄	—	4	8	no	—
18	#14519777	—	2	4	no	—
19	#14519777	—	2	2	no	—
20	#14519777	C₂×C₄	1	4	yes	—
21	C₂×C₄⋊Dic₆₁	—	2	4	no	—
22	C₄⋊#92418	D₄	1	4	yes	—
23	C₂²⋊#92418	—	2	2	no	—
24	C₄×#92418	—	2	2	no	—
25	C₂×M₁₆×C₆₁	—	2	2	no	—
26	C₂²×C₈×C₆₁	D₄	1	2	yes	—
27	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₆₁	—	2	4	no	—
29	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₆₁	D₄	1	2	yes	—
31	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	#41	1	2	yes	—
33	C₂²×Dic₆₁	—	4	2	no	—
34	C₂×Dic₁₂₂	—	2	4	no	—
35	C₂×Dic₁₂₂	—	4	8	no	—
36	C₂×Dic₁₂₂	—	4	2	no	—
37	M₁₆×C₆₁	—	4	4	no	—
38	C₂×C₈×C₆₁	—	2	4	no	—
39	C₂×C₈×C₆₁	—	4	2	no	—
40	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
41	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₆₁	C₂×D₄	1	2	yes	—
44	C₄⋊Dic₆₁	—	2	8	no	—
45	C₄⋊Dic₆₁	—	4	4	no	—
46	C₂×#92418	—	2	4	no	—
47	Dic₂₄₄	—	4	16	no	—
48	C₂³×C₆₁	#124	1	1	yes	characteristic
49	C₂×C₄×C₆₁	—	2	4	no	—
50	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
52	C₂×C₄×C₆₁	#121	1	2	yes	—
53	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
54	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
55	C₂×Dic₆₁	—	4	4	no	—
56	C₂×Dic₆₁	—	8	4	no	—
57	Dic₁₂₂	—	4	8	no	—
58	Dic₁₂₂	—	8	4	no	—
59	C₈×C₆₁	—	2	4	no	—
60	C₈×C₆₁	—	8	2	no	—
61	#92418	—	4	4	no	—
62	#9779	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	—	2	2	no	—
64	C₂²×C₆₁	#390	1	1	yes	characteristic
65	C₂²×C₆₁	#490	1	2	yes	—
66	C₄×C₆₁	—	2	2	no	—
67	C₄×C₆₁	#418	1	2	yes	—
68	C₄×C₆₁	#473	1	4	yes	—
69	C₄×C₆₁	#496	1	2	yes	—
70	Dic₆₁	—	8	4	no	—
71	C₂×#494	—	61	1	no	—
72	#1837	D₆₁	1	1	yes	characteristic
73	#1546	—	61	4	no	—
74	#1452	—	61	1	no	—
75	C₂×C₆₁	—	2	2	no	—
76	C₂×C₆₁	#1961	1	1	yes	characteristic
77	C₂×C₆₁	#1979	1	2	yes	—
78	C₂²×Q₁₆	—	122	2	no	—
79	C₂×C₄⋊Q₈	—	61	2	no	—
80	#494	—	61	4	no	—
81	#494	—	122	2	no	—
82	#431	—	61	2	no	—
83	#423	C₂×D₆₁	1	2	yes	—
84	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
85	#341	—	2	2	no	—
86	#332	—	122	8	no	—
87	#322	—	122	2	no	—
88	C₆₁	#9779	1	1	yes	characteristic, complemented, Sylow
89	C₂²×Q₈	—	122	2	no	—
90	C₂×Q₁₆	—	122	8	no	—
91	C₂×Q₁₆	—	244	8	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
94	C₄⋊Q₈	—	61	4	no	—
95	C₄⋊Q₈	—	122	2	no	—
96	C₄⋊Q₈	—	122	4	no	—
97	C₂×C₄⋊C₄	—	122	4	no	—
98	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
99	C₄⋊C₈	—	2	4	no	—
100	C₄⋊C₈	—	61	4	no	—
101	C₂²⋊C₈	—	122	2	no	—
102	C₄×C₈	—	122	2	no	—
103	C₄×C₈	C₄×D₆₁	1	2	yes	—
104	C₄×C₈	D₂₄₄	1	2	yes	—
105	C₂×Q₈	—	122	4	no	—
106	C₂×Q₈	—	244	8	no	—
107	C₂×Q₈	—	244	2	no	—
108	C₂²×C₄	—	244	2	no	—
109	C₂²×C₄	#11758226	1	2	yes	—
110	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
111	Q₁₆	—	244	16	no	—
112	M₁₆	—	4	4	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	122	4	no	—
116	C₂×C₈	#11758198	1	4	yes	—
117	C₄⋊C₄	—	122	8	no	—
118	C₄⋊C₄	—	244	4	no	—
119	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
120	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
121	C₄²	D₄×D₆₁	1	2	yes	—
122	C₂³	#14519766	1	1	yes	characteristic
123	Q₈	—	488	4	no	—
124	Q₈	—	244	8	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	488	4	no	—
127	C₂×C₄	—	244	4	no	—
128	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
129	C₂×C₄	#14519739	1	2	yes	—
130	C₂×C₄	#14519752	1	2	yes	—
131	C₂×C₄	#14519773	1	1	yes	characteristic
132	C₂×C₄	#14519802	1	4	yes	—
133	C₈	—	2	4	no	—
134	C₈	—	8	2	no	—
135	C₈	—	244	4	no	—
136	C₂²	—	2	2	no	—
137	C₂²	#16839663	1	1	yes	characteristic
138	C₂²	#16840128	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	488	4	no	—
141	C₄	#16839766	1	2	yes	—
142	C₄	#16840038	1	4	yes	—
143	C₄	#16840152	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#21446642	1	1	yes	characteristic
146	C₂	#21446732	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840142

C₂

yes

maximal, characteristic

#21444492

C₂

yes

maximal

#21444111

C₂

yes

maximal, characteristic

C₆₁×#1837

C₂

yes

maximal, characteristic

C₂²×Dic₂₄₄

—

C₂×#14519777

C₂²

yes

—

#16840142

—

#16840142

C₄

yes

—

#16839815

C₂²

yes

—

#16839509

—

#16839484

—

C₆₁×#423

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#341

—

C₂²×Dic₁₂₂

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

#14519777

—

#14519777

—

#14519777

C₂×C₄

yes

—

C₂×C₄⋊Dic₆₁

—

C₄⋊#92418

D₄

yes

—

C₂²⋊#92418

—

C₄×#92418

—

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

Dic₂₄₄

—

C₂³×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#9779

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

#390

yes

characteristic

C₂²×C₆₁

#490

yes

—

C₄×C₆₁

—

C₄×C₆₁

#418

yes

—

C₄×C₆₁

#473

yes

—

C₄×C₆₁

#496

yes

—

Dic₆₁

—

C₂×#494

—

#1837

D₆₁

yes

characteristic

#1546

—

#1452

—

C₂×C₆₁

—

C₂×C₆₁

#1961

yes

characteristic

C₂×C₆₁

#1979

yes

—

C₂²×Q₁₆

—

122

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

122

—

#431

—

#423

C₂×D₆₁

yes

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

#341

—

#332

—

122

—

#322

—

122

—

C₆₁

#9779

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

244

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₆₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

122

—

102

C₄×C₈

—

122

—

103

C₄×C₈

C₄×D₆₁

yes

—

104

C₄×C₈

D₂₄₄

yes

—

105

C₂×Q₈

—

122

—

106

C₂×Q₈

—

244

—

107

C₂×Q₈

—

244

—

108

C₂²×C₄

—

244

—

109

C₂²×C₄

#11758226

yes

—

110

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

111

Q₁₆

—

244

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

122

—

116

C₂×C₈

#11758198

yes

—

117

C₄⋊C₄

—

122

—

118

C₄⋊C₄

—

244

—

119

C₄²

C₂×C₄×D₆₁

yes

characteristic

120

C₄²

C₂×D₂₄₄

yes

characteristic

121

C₄²

D₄×D₆₁

yes

—

122

C₂³

#14519766

yes

characteristic

123

Q₈

—

488

—

124

Q₈

—

244

—

125

C₂×C₄

—

126

C₂×C₄

—

488

—

127

C₂×C₄

—

244

—

128

C₂×C₄

C₂×#11758198

yes

characteristic

129

C₂×C₄

#14519739

yes

—

130

C₂×C₄

#14519752

yes

—

131

C₂×C₄

#14519773

yes

characteristic

132

C₂×C₄

#14519802

yes

—

133

C₈

—

134

C₈

—

135

C₈

—

244

—

136

C₂²

—

137

C₂²

#16839663

yes

characteristic

138

C₂²

#16840128

yes

—

139

C₄

—

140

C₄

—

488

—

141

C₄

#16839766

yes

—

142

C₄

#16840038

yes

—

143

C₄

#16840152

yes

—

144

C₂

—

145

C₂

#21446642

yes

characteristic

146

C₂

#21446732

yes

—

	subgroup	G/H
center	C₂²	#16839663
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1837	C₂
socle	C₂²×C₆₁	#390
prime radical	C₂³×C₆₁	#124
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1837	D₆₁
hypocenter	C₆₁	#9779
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#16839663

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1837

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
88. C₆₁	#9779	semidirect

H ≅ G/N

type

88. C₆₁

#9779

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16840142	2
#21444492	2
#21444111	2
C₆₁×#1837	2
#9779	61

subgroup

index

C₆₁×#1837

Subgroups — #29579512

Conjugacy classes of subgroups

Complements

Maximal subgroups